Question

if x minus 3 and X - 1 upon 3 or both factors of a x square + 5 x + b show that a is equal to b

Answer 1

Q : If
$(x - 3) \: and \: (x - \frac{1}{3} )$
are factors of
$a {x}^{2} + 5x + b$
Prove a = b


Steps:
1) If
$(x - 3) \: \: and \: \: (x - \frac{1}{3} )$
are factors of a quadratic expression then that expression is given by

$f(x) = k(x - 3)(x - \frac{1}{3} )$
where k is real number.

$f(x) = k( {x}^{2} - ( 3 + \frac{1}{3})x + 3 \times \frac{1}{3} ) \\ f(x) = k( {x}^{2} - \frac{10x}{3} + 1) \\ f(x) = \frac{k}{3} (3 {x}^{2} - 10x + 3) \\ = \frac{2k}{3} ( \frac{3}{2} {x}^{2} - 5x + \frac{3}{2} )$

To match it with given quadratic expression,
2k/3 = 1

Hence,
We have
$\boxed {f(x) = \frac{3}{2} {x}^{2} - 5x + \frac{3}{2} }$
Here,
$a = b = \frac{3}{2}$
Hence, Proved

Answer 2

by the following picture you can do it